Which Modal Logic Is the Right One?

The question, “Which modal logic is the right one for logical necessity?,” divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Słupecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, and a more speculative argument for the claim that it does not include S4.2 is also presented. 1 The question Which if any of the many systems of modal logic in the literature is it whose theorems are all and only the right general laws of necessity? That depends on what kind of necessity is in question, so I should begin by making distinctions. A first distinction that must be noted is between metaphysical necessity or inevitability, ‘what could not have been otherwise’, and logical necessity or tautology, ‘what it is self-contradictory to say is otherwise’. The stock example to distinguish the two is this: ‘Water is a compound and not an element.’ Water could not have been anything other than what it is, a compound of hydrogen and oxygen; but there is no self-contradiction in saying, as was often said, that water is one of four elements along with earth and air and fire. The logic of inevitability might be called mood logic, by analogy with tense logic. For the one aims to do for the distinction between the indicative ‘it is the case that . . . ’ and the subjunctive ‘it could have been the case that . . . ’ something like what the other does for the distinction between the present ‘it is the case that . . . ’ and the future ‘it will be the case that . . . ’ or the past ‘it was the case that . . . ’. The logic of tautology might be called endometalogic, since it attempts to treat within the object-language notions that classical logic treats only in the metalanguage. However, it hardly deserves a name, since it immediately splits up into two subjects. For a second distinction must be made between two senses of tautology. On the one hand, there is model-theoretic logical necessity or validity, the nonexistence of Received April 16, 1998